Probability and the Sampling Distribution Quantitative Methods in HPELS 440:210

Agenda Introduction Distribution of Sample Means Probability and the Distribution of Sample Means Inferential Statistics

Introduction Recall:  Any raw score can be converted to a Z-score Provides location relative to µ and   Assuming NORMAL distribution: Proportion relative to Z-score can be determined Z-score relative to proportion can be determined  Previous examples have looked at single data points  Reality  most research collects SAMPLES of multiple data points Next step  convert sample mean into a Z- score  Why? Answer probability questions

Introduction Two potential problems with samples: 1. Sampling error  Difference between sample and parameter 2. Variation between samples  Difference between samples from same taken from same population  How do these two problems relate?

Agenda Introduction Distribution of Sample Means Probability and the Distribution of Sample Means Inferential Statistics

Distribution of Sample Means Distribution of sample means = sampling distribution is the distribution that would occur if:  Infinite samples were taken from same population  The µ of each sample were plotted on a FDG Properties:  Normally distributed  µ M = the “mean of the means”   M = the “SD of the means” Figure 7.1, p 202

Distribution of Sample Means Sampling error and Variation of Samples Assume you took an infinite number of samples from a population  What would you expect to happen?  Example 7.1, p 203

Assume a population consists of 4 scores (2, 4, 6, 8) Collect an infinite number of samples (n=2)

Total possible outcomes: 16 p(2) = 1/16 = 6.25%p(3) = 2/16 = 12.5% p(4) = 3/16 = 18.75%p(5) = 4/16 = 25% p(6) = 3/16 = 18.75%p(7) = 2/16 = 12.5% p(8) = 1/16 = 6.25%

Central Limit Theorem For any population with µ and , the sampling distribution for any sample size (n) will have a mean of µ M and a standard deviation of  M, and will approach a normal distribution as the sample size (n) approaches infinity If it is NORMAL, it is PREDICTABLE!

Central Limit Theorem The CLT describes ANY sampling distribution in regards to: 1. Shape 2. Central Tendency 3. Variability

Central Limit Theorem: Shape All sampling distributions tend to be normal Sampling distributions are normal when:  The population is normal or,  Sample size (n) is large (>30)

Central Limit Theorem: Central Tendency The average value of all possible sample means is EXACTLY EQUAL to the true population mean  µ M = µ If all possible samples cannot be collected?  µ M approaches µ as the number of samples approaches infinity

µ = / 4 µ = 5 µ M = / 16 µ M = 80 / 16 = 5

Central Limit Theorem: Variability The standard deviation of all sample means is denoted as  M  M =  /√n Also known as the STANDARD ERROR of the MEAN (SEM)

SEM  Measures how well statistic estimates the parameter  The amount of sampling error between M and µ that is reasonable to expect by chance Central Limit Theorem: Variability

SEM decreases when:  Population  decreases  Sample size increases Other properties:  When n=1,  M =  (Table 7.2, p 209)  As SEM decreases the sampling distribution “tightens” (Figure 7.7, p 215)  M =  /√n

Agenda Introduction Distribution of Sample Means Probability and the Distribution of Sample Means Inferential Statistics

Probability  Sampling Distribution Recall:  A sampling distribution is NORMAL and represents ALL POSSIBLE sampling outcomes  Therefore PROBABILITY QUESTIONS can be answered about the sample relative to the population

Probability  Sampling Distribution Example 7.2, p 209 Assume the following about SAT scores:  µ = 500   = 100  n = 25  Population  normal What is the probability that the sample mean will be greater than 540? Process: 1. Draw a sketch 2. Calculate SEM 3. Calculate Z-score 4. Locate probability in normal table

Step 1: Draw a sketch Step 2: Calculate SEM SEM =  M =  /√n SEM = 100/√25 SEM = 20 Step 3: Calculate Z-score Z = 540 – 500 / 20 Z = 40 / 20 Z = 2.0 Step 4: Probability Column C p(Z = 2.0) =

Agenda Introduction Distribution of Sample Means Probability and the Distribution of Sample Means Inferential Statistics

Looking Ahead to Inferential Statistics Review:  Single raw score  Z-score  probability Body or tail  Sample mean  Z-score  probability Body or tail What’s next?  Comparison of means  experimental method

Textbook Assignment Problems: 13, 17, 25 In your words, explain the concept of a sampling distribution In your words, explain the concept of the Central Limit Theorum